Geodesic
A note on the number of edges guaranteeing a \(C_4\) in Eulerian bipartite digraphs
The electronic journal of combinatorics, Tome 9 (2002)
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Let $G$ be an Eulerian bipartite digraph with vertex partition sizes $m,n$. We prove the following Turán-type result: If $e(G) > 2mn/3$ then $G$ contains a directed cycle of length at most 4. The result is sharp. We also show that if $e(G)=2mn/3$ and no directed cycle of length at most 4 exists, then $G$ must be biregular. We apply this result in order to obtain an improved upper bound for the diameter of interchange graphs.
DOI : 10.37236/1667
Classification : 05C20, 05C35, 05C45
Mots-clés : digraph, extremal theory, cycle 
@article{10_37236_1667,
     author = {Jian Shen and Raphael Yuster},
     title = {A note on the number of edges guaranteeing a {\(C_4\)} in {Eulerian} bipartite digraphs},
     journal = {The electronic journal of combinatorics},
     year = {2002},
     volume = {9},
     doi = {10.37236/1667},
     zbl = {0994.05070},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1667/}
}
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Jian Shen; Raphael Yuster. A note on the number of edges guaranteeing a \(C_4\) in Eulerian bipartite digraphs. The electronic journal of combinatorics, Tome 9 (2002). doi: 10.37236/1667

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